Average Error: 15.0 → 0.3
Time: 9.0s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{\sin \left(x \cdot 0.5\right)}{\sin x} \cdot \frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}\]
\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}
↓
\frac{\sin \left(x \cdot 0.5\right)}{\sin x} \cdot \frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}
double f(double x) {
        double r687920 = 8.0;
        double r687921 = 3.0;
        double r687922 = r687920 / r687921;
        double r687923 = x;
        double r687924 = 0.5;
        double r687925 = r687923 * r687924;
        double r687926 = sin(r687925);
        double r687927 = r687922 * r687926;
        double r687928 = r687927 * r687926;
        double r687929 = sin(r687923);
        double r687930 = r687928 / r687929;
        return r687930;
}
↓
double f(double x) {
        double r687931 = x;
        double r687932 = 0.5;
        double r687933 = r687931 * r687932;
        double r687934 = sin(r687933);
        double r687935 = sin(r687931);
        double r687936 = r687934 / r687935;
        double r687937 = 8.0;
        double r687938 = r687937 * r687934;
        double r687939 = 3.0;
        double r687940 = r687938 / r687939;
        double r687941 = r687936 * r687940;
        return r687941;
}

Error

Bits error versus x

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Results

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Target

Original15.0
Target0.3
Herbie0.3
\[\frac{\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}\]

Derivation

  1. Initial program 15.0

    \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity15.0

    \[\leadsto \frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\color{blue}{1 \cdot \sin x}}\]

  4. Applied times-frac0.5

    \[\leadsto \color{blue}{\frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{1} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}}\]

  5. Simplified0.5

    \[\leadsto \color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]
  6. Using strategy rm

  7. Applied pow10.5

    \[\leadsto \left(\frac{8}{3} \cdot \color{blue}{{\left(\sin \left(x \cdot 0.5\right)\right)}^{1}}\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]

  8. Applied pow10.5

    \[\leadsto \left(\color{blue}{{\left(\frac{8}{3}\right)}^{1}} \cdot {\left(\sin \left(x \cdot 0.5\right)\right)}^{1}\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]

  9. Applied pow-prod-down0.5

    \[\leadsto \color{blue}{{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}^{1}} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]

  10. Simplified0.3

    \[\leadsto {\color{blue}{\left(\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}\right)}}^{1} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]

  11. Final simplification0.3

    \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\sin x} \cdot \frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

  :herbie-target
  (/ (/ (* 8 (sin (* x 0.5))) 3) (/ (sin x) (sin (* x 0.5))))

  (/ (* (* (/ 8 3) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))